N u m e ric al s t u dy of slip effec t s on u n s t e a dy ays m m e t ric

bioconvec tive n a nofluid flow in a po ro u s mic roc h a n n el wi th a n exp a n din g/ con t r a c ting u p p e r

w all u sin g Buon gio r no’s m o d elBe g, OA, Basir, M F M, U d din, MJ a n d Ism ail, AIM

h t t p://dx.doi.o r g/10.1 1 4 2/S 0 2 1 9 5 1 9 4 1 7 5 0 0 5 9 2

Tit l e N u m e ric al s t u dy of slip effec t s on u n s t e a dy ays m m e t ric bioconvec tive n a nofluid flow in a po rou s mic roc h a n n el wi th a n exp a n ding/ con t r ac tin g u p p e r w all u sing Buon gio rno’s m o d el

Aut h or s Beg, OA, Basir, M F M, U d din, MJ a n d Is m ail, AIM

Typ e Article

U RL This ve r sion is available a t : h t t p://usir.s alfor d. ac.uk/id/e p rin t/40 1 0 0/

P u bl i s h e d D a t e 2 0 1 6

U SIR is a digi t al collec tion of t h e r e s e a r c h ou t p u t of t h e U nive r si ty of S alford. Whe r e copyrigh t p e r mi t s, full t ex t m a t e ri al h eld in t h e r e posi to ry is m a d e fre ely availabl e online a n d c a n b e r e a d , dow nloa d e d a n d copied for no n-co m m e rcial p riva t e s t u dy o r r e s e a r c h p u r pos e s . Ple a s e c h e ck t h e m a n u sc rip t for a ny fu r t h e r copyrig h t r e s t ric tions.

For m o r e info r m a tion, including ou r policy a n d s u b mission p roc e d u r e , ple a s e

con t ac t t h e Re posi to ry Tea m a t : u si r@s alford. ac.uk .

1

JOURNAL OF MECHANICS IN MEDICINE AND BIOLOGY

ARTICLE ID - JMMB-D-16-00106

Print ISSN: 0219-5194 Online ISSN: 1793-6810

IMPACT FACTOR = 0.797 PUBLISHER: WORLD SCIENTIFIC (SINGAPORE)

Accepted September 7th

2016

NUMERICAL STUDY OF SLIP EFFECTS ON UNSTEADY AYSMMETRIC

BIOCONVECTIVE NANOFLUID FLOW IN A POROUS MICROCHANNEL WITH AN

EXPANDING/ CONTRACTING UPPER WALL USING BUONGIORNO’S MODEL

O. Anwar Bég1*

, Md. Faisal Md Basir2*

, M.J. Uddin3 and A. I. Md. Ismail

2

1 Fluid Mechanics, Biomechanics and Propulsion, Aeronautical and Mechanical Engineering

Division, Room UG17, Newton Building, University of Salford, M54WT, UK. 2

School of Mathematical Sciences, Universiti Sains Malaysia,11800, Penang, Malaysia. 3American International University-Bangladesh, Banani, Dhaka 1213, Bangladesh.

*Corresponding author: Email:O.A.Beg@salford.ac.uk; gortoab@gmail.com

ABSTRACT

In this paper, the unsteady fully developed forced convective flow of viscous incompressible

biofluid that contains both nanoparticles and gyrotactic microorganisms in a horizontal

micro-channel is studied. Buongiorno’s model is employed. The upper channel wall is either

expanding or contracting and permeable and the lower wall is static and impermeable. The

plate separation is therefore a function of time. Velocity, temperature, nano-particle species

(mass) and motile micro-organism slip effects are taken into account at the upper wall. By

using the appropriate similarity transformation for the velocity, temperature, nanoparticle

volume fraction and motile microorganism density, the governing partial differential

conservation equations are reduced to a set of similarity ordinary differential equations.

These equations under prescribed boundary conditions are solved numerically using the

Runge-Kutta-Fehlberg fourth-fifth order numerical quadrature in the MAPLE symbolic

software. Excellent agreement between the present computations and solutions available in

the literature (for special cases) is achieved. The key thermofluid parameters emerging are

identified as Reynolds number, wall expansion ratio, Prandtl number, Brownian motion

parameter, thermophoresis parameter, Lewis number, bioconvection Lewis number and

bioconvection Péclet number. The influence of all these parameters on flow velocity,

temperature, nano-particle volume fraction (concentration) and motile micro-organism

density function is elaborated. Furthermore graphical solutions are included for skin friction,

wall heat transfer rate, nano-particle mass transfer rate and micro-organism transfer rate.

Increasing expansion ratio is observed to enhance temperatures and motile micro-organism

density. Both nanoparticle volume fraction and microorganism increases with an increase in

momentum slip. The dimensionless temperature and microorganism increases as wall

expansion increases. Applications of the study arise in advanced nanomechanical

bioconvection energy conversion devices, bio-nano-coolant deployment systems etc.

Keywords: Multiple slip, bioconvection, nanofluids, unsteady flow; micro-channel; porous

wall; gyrotactic microorganisms; expanding/contracting upper wall; numerical.

2

NOMENCLATURE

a

length of channel m

a

time-dependent rate m

s

1a

velocity slip parameter,1

1

Sa

a

A

injection coefficient m

s

b

chemotaxis constant m

1b

thermal slip parameter,1

1

Db

a

C nano-particles volume fraction

0C reference nano-particle volume fraction

1C nano-particle volume fraction on lower wall

2C nano-particle volume fraction on lower wall

1c

mass slip parameter,1

1

Ec

a

pc specific heat at constant pressure J

kgK

BD Brownian diffusion coefficient

2m

s

nD variable micro-organism diffusion coefficient

2m

s

TD thermophoretic diffusion coefficient

2m

s

1D thermal slip factor m

1d

micro-organism slip parameter,1

1

Fd

a

1E mass slip factor m

1F micro-organism slip factor m

( )f dimensionless stream function

3

j vector flux of micro-organisms 2

kg

m s

k thermal conductivity W

mK

Nb Brownian motion parameter 2 0

0

( )BD C C

Nb

Nt thermophoresis parameter 2 0

0 0

( )TD T T

NtT

N number of motile micro-organisms

1N lower wall motile micro-organisms

2N upper wall motile micro-organisms

p pressure 2

( )Newton

m

Pe bioconvection Péclet number ( )c

n

bWPe

D

Pr Prandtl number 0

Pr ( )

Re Reynolds number wav

\

1S velocity slip factor s

m

Lb bioconvection Lewis number 0 ( )n

SbD

Le Lewis number 0 ( )B

LeD

t dimensional time ( )s

T nanofluid temperature ( )K

0T reference temperature ( )K

1T temperature on lower wall ( )K

2T temperature on upper wall ( )K

u velocity components along the x axis m

s

v velocity vector m

s

v

average swimming velocity vector of micro-organism 2m

s

4

v velocity components along the y axis m

s

wv dimensional inflow/outflow velocity m

s

cW maximum cell swimming speed m

s

x dimensional coordinate along the surface m

y coordinate normal to the surface m

Greek letters

0 effective thermal diffusivity

2m

s

wall expansion ratio aa

independent similarity variable

( ) dimensionless temperature

1 constant

kinematic viscosity

2m

s

fluid density 3

kg

m

f

c volumetric heat capacity of the fluid 3

J

m K

p

c volumetric heat capacity of the nanoparticle material 3

J

m K

ratio of the effective heat capacity of the nanoparticle material to the fluid heat

capacity ( )

( )

p

f

c

c

( ) dimensionless nanoparticles volume fraction

1 constant

( ) dimensionless number of motile micro-organisms

1 constant

Subscripts

( ) ' ordinary differentiation with respect to

5

0( ) condition at reference

1( ) condition at lower wall

2( ) condition at upper wall

1. INTRODUCTION

The recent progress in scaling down devices in medical engineering requires a more

elegant and refined understanding of fluid dynamic, heat and mass transfer phenomena at

small scales. Responding to this demand, numerous scientists and engineers are actively

studying methods for enhancing heat, mass and momentum transfer strategies at the micro-

dimension scale. Usually channels with hydraulic diameter below 1 mm are categorized as

micro-channels. They arise in diverse areas of health technology including efficient micro-

sized cooling systems for biomedical processing systems, bio-electronic devices, lab-on-a-

chip biological designs, bio-astronautics, cell sorting, immunoassays, DNA sampling etc [1].

One aspect distinguishing micro/nanoscale gas flows from their macroscale counterparts is

that the wall slip effect usually becomes so vital at micro/nanoscales that its negligence may

lead to predictions deviating unacceptably from physical reality. Employing the Navier–

Stokes equations in conjunction with adequate slip velocity boundary conditions that properly

incorporate gas molecule and wall interaction kinetics, has been demonstrated to be a robust

methodology for providing accurate results for micro/nanoscale gas flows as described by

Wu [2]. Numerous studies have revealed the important influence of slip on near-wall flow

characteristics.

The internal flow through a porous channel with extending/contracting walls has

some important applications in biophysical flows such as blood flow and artificial dialysis,

air and blood circulation in the respiratory system. Further applications include filtration in

tissue, oxygen diffusion in capillaries, cosmetics materials manufacture, the mechanics of the

cochlea in the human ear, pulsating diaphragms and cerebral hydrodynamics [3, 4]. A

significant number of investigations have been communicated recently on internal fluid

dynamics of porous channels under different wall conditions and for various transport

phenomena. Adesanya [5] examined natural convective flow of heat generating fluid through

a permeable channel with anisotropic slip. A seminal early analysis was presented by Uchida

and Aoki [6] concerning viscous flow through a tube with a contracting cross section. Later

Bujurke [7] further considered the Uchida-Aoki model both analytically with series

expansions and numerically. Goto and Uchida [8] presented an unsteady laminar flow model

for incompressible fluid through a permeable pipe. Other recent studies focused on fluid

6

mechanics in a porous channel with a contracting and/or expanding wall include the works by

Hatami et al. [9], Boutros et al. [10], Xinhui et al. [11], Si et al. [12], Ahmed et al. [13] and

Darvishi et al. [14]. These articles have considered many multi-physical effects including

non-Newtonian fluids, magnetohydrodynamics, chemical reaction and heat and mass transfer.

Nanofluids also constitute a significant development in thermofluid dynamics in

recent years. The term nanofluid was popularized by Choi [15]. A nanofluid is characterized

as the suspension synthesized via scattering of nano-sized particles in a base fluid. With the

rapid develops in nano-manufacturing, many low-cost combinations of liquid/particle are

now obtainable. These include particles of metals such aluminum, copper, gold, iron and

titanium or their oxides. The base fluids used are usually water, ethylene glycol, toluene and

oil. A key feature of nanofluids is their demonstrated ability to improve the efficiency of heat

transfer equipment as documented by Eastman et al. [16]. Khairul et al. [17] reported on

experimental results concerning the impressive thermal properties of the nanofluids. The

acceleration in deployment of nanofluids has required both extensive laboratory and field

testing and also theoretical and computational simulations. The fundamental idea of

nanofluids is to combine the two substances to form a heat transfer medium that behaves like

a fluid, but has the high thermal conductivity characteristics of a metal. Nanofluids have

therefore been deployed in a wide spectrum of technological applications including

microelectronics, solar cells, pharmaceutical processes, hybrid-powered engines, vehicle

thermal management, refrigeration/chiller systems, heat exchangers, nuclear reactor coolants,

lubricants, space technology, combustion devices and rocket propellants, as elaborated by

Bég and Tripathi [18]. Extensive reviews and analyses of nanofluids have been reported by

many researchers – see for example Kakaç and Pramuanjaroenkij [19], Hamad and Ferdows

[20].

A substantial thrust in modern fluid mechanics has been biological transport. A sub-

set of such flows is bioconvection. Bioconvection can be defined as pattern formation in

suspensions of microorganisms, such as bacteria and algae, due to up-swimming of the

microorganisms towards a specific taxis e.g. light, gravity, chemicals, magnetic fields,

oxygen. Detailed elaboration is given in Pedley et al. [21]. It results from an unstable density

stratification induced by upswimming microorganisms which occurs when the

microorganisms (heavier than water) accumulate in the upper regions of the fluid,

manifesting in a special type of hydrodynamic instability which are observable as

bioconvection plumes, as discussed by Uddin et al. [22]. A refinement in bioconvection is

7

achieved by the careful suspension of microorganisms in nanofluids which serves to enhance

thermal conductivity as well as stability of the resulting bio-nanofluid. Raees et al. [23]

studied computationally the forced/free convection in buoyancy-driven nano-liquid film flow

containing gyrotactic microorganisms using both a passively and actively controlled

nanofluid with a finite difference technique. Raees et al. [24] simulated numerically the

unsteady squeezing flow and heat transfer through a channel in a nanofluid containing

microorganisms with wall expansion and contraction effects. Basir et al. [25] investigated

transient multi-slip nanofluid bioconvection from an extending cylindrical body and

considered the effects of hydrodynamic, thermal and micro-organism slip.

Motivated by further refining physico-mathematical modelling of channel bio-

convection nanofluid flows, in the present work, we extend the analysis of Xinhui et al. [11]

to obtain solutions of the flow, heat, nano-particle and micro-organism diffusion in unsteady

bioconvective nanofluid flow in a porous channel with an expanding and contracting wall.

Here we considered four slip mechanisms i.e. velocity, thermal, mass (nano-particle species)

and micro-organism slip at the upper wall of a two-dimensional channel. The channel plates

are parallel, the upper plate being porous and the lower plate being impermeable. Validation

of the solution obtained is achieved with an excellent agreement.

2. MATHEMATICAL MODEL

Consider a two-dimensional unsteady laminar forced bioconvective slip flow of nanofluid in

a horizontal semi-infinite channel with the distance a t between two walls (see Fig. 1).

The upper channel wall has nanofluid injected at velocity, wv , and is expanding or

contracting uniformly at a time-dependent rate, a

t

. The plate separation is therefore a

function of time and equals ).(ta The lower channel wall is a static impermeable surface.

The coordinate system is selected such that the x axis is along the lower plate and the

y axis is normal to the plate. u and v denote the velocity components in the x and y

directions, respectively. , ,T C N are the temperature of the fluid, the nanoparticle volume

fraction and density of motile micro-organisms, respectively. The temperatures are assumed

to be constant on the lower and upper walls and are denoted, respectively by as 1T and 2T .

In addition to this, the nanoparticle volume fraction on the upper wall is 2C and that on the

8

lower wall is 1C . The density of motile microorganisms on the upper wall is prescribed as

2N and on the lower wall as 1N .

2 2,slip slipT T T C C C

2 slipN N N

slipu u at y = a t ,wv Aav

a t

1 1,T T C C

1N N

Figure 1: Physical configuration for bioconvection nanofluid slip channel flow with an

expanding or contracting wall (model for adaptive microbial nanofluid fuel cell)

Here the base fluid is taken as water and nano-particles are suspended in the base fluid. The

resulting nanofluid is assumed to be stable and does not permit agglomeration in the fluid,

thereby allowing the micro-organisms to be alive [24]. Also the direction of microorganism’s

swimming is independent of nanoparticles [25]. Under these assumptions we employ the

Buongiorno nanofluid model [26] which emphasizes Brownian motion and thermophoresis as

the principal mechanisms for nano-particle thermal enhancement. This model generally

assumes that the nanofluid is dilute and contains a homogenous suspension of equally-sized,

nano-particles in thermal equilibrium. It infers that thermophoresis and Brownian motion

effects dominate over other nanoscale effects such as ballistic collisions, micro-convection

patterns etc. The Buongiorno model has been proven to widely applicable in nanoscale

transport phenomena and extensive investigations have been reported. These include

Malvandi et al. [27] for buoyancy-driven pumping, Xu et al. [28] for mixed convection,

Sheremet et al. [29] for nanofluid flows in porous media enclosures, Akbar et al. [30] for

electro-conductive nanofluid extrusion, Uddin et al. [31] for high temperature radiative

processing of aerospace nanomaterials and also very recently by Uddin et al. [32] for

anisotropic slip stagnation nanofluid gyrotactic bioconvection flows. Buongiorno’s model

O x

y

Expanding/contracting upper wall

Impermeable static lower wall

Gyrotactic (compensating torque)

driven micro-organisms

Metallic nano-particles

Wall mass flux (suction/injection)

Lower wall conditions

Upper wall conditions

9

was originally adapted for laminar boundary layers by Kuznetsov and Nield [33]. Using this

model, the governing partial differential equations of mass, momentum, thermal energy,

nano-particle concentration and micro-organism density conservation emerge in the

following forms:

0,v (1)

21,

vv v p v

t

(2)

2

0

0

TB

DTv T T D T C T T

t T

, (3)

2 2

0

TB

DCv C D C T

t T

, (4)

0n

jt

(5)

Here ,v u v are the velocity components along and perpendicular to the wall, p is the

pressure, ( )

( )

p

f

c

c

is the ratio of nano-particle heat capacity and the base fluid heat

capacity, 0( ) f

k

c

is the thermal diffusivity of the fluid, is the density of the base fluid,

is the kinematic viscosity, BD is the Brownian diffusion coefficient and TD is the

thermophoresis diffusion coefficient, 0T is the reference temperature. Furthermore, j which

denotes the flux of microorganisms can be written as:

nj N v N v D N

(6)

Also 2 0

cv CC C

bW

is the average swimming velocity vector of micro-organisms in

nanofluids, nD designates the species diffusivity of micro-organisms, b denotes the

chemotaxis constant, Wc signifies the maximum cell swimming speed, 0C is the reference

concentration and C

Cy

. For the two-dimensional case, Eqns. (2)–(5) can be written as:

0,u v

x y

(7)

10

2 2

2 2

1,

u u u p u uu v

t x y x x y

(8)

2 2

2 2

1,

v v u p v vu v

t x y y x y

(9)

2 2

0 2

0

2

22

,TB

T T T T Tu v

t x y x y

DT C T C T TD

x x y y T x y

(10)

2 2 2 2

2 2 2 2

0

,TB

DC C Cu v D

t

C C

x y x y T x

C

y

C

(11)

2

2 2

0

2 2

.ncbWN N N C C N N

u v N N Dt x y C x x y y x yC

(12)

The relevant boundary conditions are following Dauenhauer and Majdalani [34], Cao et al.

[35] and Raza et al. [36]:

2 2 2

1 1 1

, ,

, at    

0, 0, , at     0

,

,

slip

slip slip slip

wu u v

T T T C C C N N N y a t

u v T T C C N N y

v Aa t

(13)

where 1slip

uu S

y

is the linear slip velocity, 1S is the velocity slip factor, 1slipT DT

y

is

the thermal slip, 1D is the thermal slip factor, 1slipC EC

y

is the mass (nano-particle) slip,

1E is the mass (nano-particle) slip factor, 1slipN FN

y

is the micro-organism slip and 1F is

the micro-organism slip factor. wA v a t is the coefficient of the injection which is a

measure of permeability of the wall [34]. Defining the following similarity transformations

[11]:

11

2

0 0

2 0 2 0 2

, ( , ) , , ,

, , .

yu x F t v t

a t a t

T T C C N

T T C

t

C N

Fa

(14)

By substituting Eqns. (14) into Eqns. (7) – (12) together with boundary conditions (13) and

eliminating the pressure gradient terms from the momentum equations (8) and (9), the

resulting system of similarity differential equations with respect to both time and space can

be obtained following transformation [6], [11]: as follows:

0)3()Re( /////////// fffffff iv (15)

2Pr Re Pr 0f Nb Nt (16)

Pr Re Pr 0Nt

Le f LeNb

(17)

Pr Re Pr 0Pe Lb f Lb (18)

where Re

Ff , Re wav

is the permeation Reynolds number,

a a

is the wall expansion

ratio, 0

Pr

is the Prandtl number,

2 0

0

BD C CNb

is the Brownian motion

parameter, 2 0

0 0

TD T TNt

T

is the thermophoresis parameter, 0

B

LeD

is the Lewis

number, 0

n

LbD

is the bioconvection Lewis number and c

n

bWPe

D is the bioconvection

Péclet number. Note that 0 corresponds to an expanding upper wall and

0 corresponds to a contracting upper wall. For =0 the upper wall is static i.e. the

channel plate separation is fixed.

The corresponding boundary conditions are:

1 1 1

1 1 1 1

0 0, 0 0, 0 , 0 , 0 ,

1 1, 1 1 , 1 1 1 , 1 1 1 , 1 1 1

f f

f f a f b c d

(19)

12

Where 11

Sa

a

is the velocity slip parameter, 1

1

Db

a is the thermal slip parameter,

11

Ec

a is the mass (nano-particle) slip parameter, 1

1

Fd

a is the microorganism slip

parameter, and 1 0 1 0 11 1 1

2 0 2 0 2

, ,T T C C N

T T C C N

are constants. In the present work we

confine attention to positive permeation Reynolds number which physically corresponds to

injection at the upper wall i.e. suction (negative permeation Reynolds number is not

considered since it is not relevant to coolant designs). The model developed herein considers

viscous, incompressible, biological fluid suspensions. Many such fluids exist including water-

based liquids containing bacterial (Bacillus, Chlamydomonas, Volvox, and Tetrahymena) ,

algal or zooplankton suspensions, as elaborated by Pedley and Kessler [37]. The presence of

nano-particles does not alter the incompressibility or biological nature of such suspensions as

the nano-particles do not interact with the motile (i.e. self-propelled) micro-organisms. They

are separate “species”. The micro-organisms possess mean diameters which range from 1 to

200 micro-metres and are therefore significantly larger than nano-particle dimensions. These

micro-organisms, which are considered to be gyrotactic in the present study, possess slightly

greater densities than the water base fluid e.g. several percent for algae and no more than 10

percent for bacteria e.g. B. subtilis. Excellent details are provided by Hart & Edwards [38].

3. PHYSICAL QUANTITIES FOR MEDICAL ENGINEERING DESIGN

The quantities of practical interest in this study are the Nusselt number xNu ,

Sherwood number ,xSh and the density number of motile micro-organisms xNn , which are

defined, respectively as:

2 0 2 0 2

, , .

( ) (C )

w m n

x x x

B n

a t q a t q a t qNu Sh Nn

k T T D C D N

(20)

where , and w m nq q q represent the surface heat flux, surface mass flux and the surface

motile microorganism flux, respectively and are defined by:

, , .w m B n n

y a t y a t y a t

T C Nq k q D q D

y y y

(21)

Substituting Eqs. (14), (21) into (20), the following expressions are obtained:

13

'(1) , '(1) , '(1) .x x xNu Sh Nn (22)

4. NUMERICAL SOLUTION AND VALIDATION

The two point boundary problem defined by Eqns. (15) – (18) under boundary conditions

(19) is strongly nonlinear. A computational method is therefore adopted, namely the Runge-

Kutta-Felhberg fourth-fifth order shooting method available in the Maple software via built-

in functions. This method has been successfully used by many researchers in order to solve

complex higher order, nonlinear ordinary differential equations (ODEs). The details of the

algorithm are lucidly documented in Uddin et al. [39], Bég et al. [40] and Bég and Makinde

[41] and are therefore not repeated here. Validation of the general model developed is not

possible with published solutions from the literature and therefore we employ a second order

accurate finite difference algorithm known as Nakamura’s method to validate the general

Maple solutions. The Nakamura tridiagonal method [42] generally achieves fast convergence

for nonlinear viscous flows which may be described by either parabolic (boundary layer) or

elliptic (Navier-Stokes) equations. The coupled 10th

order system of nonlinear, multi-degree,

ordinary differential equations defined by (15)–(18) with boundary conditions (19) is solved

using the NANONAK code in double precision arithmetic in Fortran 90, as elaborated by

Bég [43]. Computations are performed on an SGI Octane Desk workstation with dual

processors and take seconds for compilation. As with other difference schemes, a reduction in

the higher order differential equations, is also fundamental to Nakamura’s method. The

method has been employed successfully to simulate many sophisticated nonlinear transport

phenomena problems e.g. magnetized bio-rheological coating flows (Bég et al. [44]). Earlier

it has been employed in viscoelastic Falkner-Skan flows (Bég et al. [45] and rotating

micropolar convection flows (Gorla and Nakamura [46]). Intrinsic to this method is the

discretization of the flow regime using an equi-spaced finite difference mesh in the

transformed coordinate (). The partial derivatives for f, , and with respect to are

evaluated by central difference approximations. An iteration loop based on the method of

successive substitution is utilized to advance the solution i.e. march along. The finite

difference discretized equations are solved in a step-by-step fashion on the -domain. For the

energy, nano-particle species and motile micro-organism density conservation Eqns. (16) -

(18) which are second order multi-degree ordinary differential equations, only a direct

substitution is needed. However a reduction is required for the fourth order momentum Eqn.

(25). We apply the following substitutions:

14

P = f ///

(23)

Q = (24)

R = (25)

S = (26)

The ODEs (15)-(18) then retract to:

Nakamura momentum equation:

11

/

1

//

1 TPCPBPA (27)

Nakamura energy equation:

22

/

2

//

2 TQCQBQA (28)

Nakamura nano-particle species equation:

33

/

3

//

3 TRCRBRA (29)

Nakamura motile micro-organism density number equation:

44

/

4

//

4 TSCSBSA (30)

Here Ai=1,2,3,4, Bi=1,2,3,4, Ci=1,2,3,4 are the Nakamura matrix coefficients, Ti=1,2,3,4 are

the Nakamura source terms containing a mixture of variables and derivatives associated with

the respective lead variable (P, Q, R, S). The Nakamura Eqns. (27)–(30) are transformed to

finite difference equations and these are orchestrated to form a tridiagonal system which due

to the high nonlinearity of the numerous coupled, multi-degree terms in the momentum,

energy, nano-particle species and motile micro-organism density conservation equations, is

solved iteratively. Householder’s technique is ideal for this iteration. The boundary

conditions (19) are also easily transformed. Further details of the NTM approach are

provided in the comprehensive treatise of Nakamura [47]. Applications in non-Newtonian

magnetic channel flows are documented in the review by Bég [48]. Comparisons are

documented in Tables 1-4 for heat transfer rate and nano-particle mass transfer rate at the

upper wall ( =1) i.e. - /(1) and -

/(1) , respectively. Generally very close correlation

is obtained over a range of (contracting, expanding and stationary upper wall cases), b1

and c1 values. Confidence in the Maple RK45 numerical solutions is therefore justifiably

high.

15

b1 - /(1)

Maple

- /(1)

Nakamura

- /(1)

Maple

- /(1)

Nakamura

- /(1)

Maple

- /(1)

Nakamura

=-0.5 = -0.5 =0 =0 =+0.5 =+0.5

0.25 0.43851662 0.43851788 0.10479830 0.10479828 0.02415781 0.02415742

0.50 0.55356648 0.55356475 0.11035680 0.11035704 0.02444230 0.02444204

0.75 0.74039460 0.74039503 0.11650804 0.11650796 0.02473331 0.02473307

1.00 1.06979037 1.06979217 0.12334615 0.12334589 0.02503103 0.02503096

Table 1: Values of 1 1 1'(1) for 0.1, 0.5, 1Nt Nb a c d Pe Le Lb with Re =

0.5 for various expansion ratio () and thermal slip parameter (b1) values.

b1 - /(1)

Maple

- /(1)

Nakamura

- /(1)

Maple

- /(1)

Nakamura

- /(1)

Maple

- /(1)

Nakamura

=-0.5 = -0.5 =0 =0 =+0.5 =+0.5

0.25 0.30960126 0.30960098 0.07392428 0.07392387 0.01684928 0.01684864

0.50 0.36323052 0.36323104 0.07665052 0.07665002 0.01698722 0.01698691

0.75 0.43746994 0.43746887 0.07957625 0.07957597 0.01712735 0.01712702

1.00 0.54493004 0.54493028 0.08272295 0.08272316 0.01726972 0.01726958

Table 2: Values of 1 1 1'(1) for 0.1, 0.5, 1Nt Nb a c d Pe Le Lb with Re =

0.6 for various expansion ratio () and thermal slip parameter (b1) values.

c1 -/(1)

Maple

-/(1)

Nakamura

-/(1)

Maple

- /(1)

Nakamura

- /(1)

Maple

-/(1)

Nakamura

=-0.5 = -0.5 =0 =0 =+0.5 =+0.5

0.25 0.87883680 0.87883613 0.48948165 0.48948214 0.25131513 0.25131495

0.50 1.02368938 1.02368895 0.52535169 0.52535196 0.25985873 0.25985808

0.75 1.22567972 1.22567904 0.56689568 0.56689492 0.26900392 0.26900343

1.00 1.52688227 1.52688256 0.61557581 0.61557505 0.27881661 0.27881606

Table 3: Values of -/(1) for Nt = Nb = a1 = b1 = d1=0.1, Pe = Le = Lb = 0.5 with Re =

0.9 for various expansion ratio () and nano-particle mass slip parameter (c1) values.

16

c1 -/(1)

Maple

-/(1)

Nakamura

-/(1)

Maple

- /(1)

Nakamura

- /(1)

Maple

-/(1)

Nakamura

=-0.5 = -0.5 =0 =0 =+0.5 =+0.5

0.25 0.77028858 0.77028903 0.41889688 0.41889712 0.21232448 0.21232463

0.50 0.87377120 0.87377201 0.44433169 0.44433221 0.21833443 0.21833491

0.75 1.00936233 1.00936197 0.47305578 0.47305614 0.22469470 0.22469405

1.00 1.19474231 1.19474274 0.50575154 0.50575093 0.23143684 0.23143702

Table 4: Values of -/(1) for Nt = Nb = a1 = b1 = d1=0.1, Pe = Le = Lb = 0.5 with Re =

1.0 for various expansion ratio () and nano-particle mass slip parameter (c1) values.

Tables 1 -4 show that temperature gradient i.e. upper wall heat transfer rate, '(1) ,

decreases with increasing expansion parameter ratio and permeation Reynolds number.

'(1) however increases with increasing magnitude of thermal slip, 1.b Furthermore nano-

particle solutal gradient i.e. nano-particle upper wall mass transfer rate '(1) is found to

consistently reduce with increasing and Re. '(1) whereas it is enhanced significantly

with increasing nano-particle mass slip, 1.c

5. COMPUTATIONAL RESULTS AND DISCUSSION

In this section Maple numerical graphical solutions are presented for the effects of several

controlling parameters on the dimensionless velocity, f , temperature , nanoparticle

volume fraction and motile micro-organism density function . We consider water-

based bio-nanofluid for which Pr = 6.8. Maple solutions are presented in Figs. 2-10.

Figure 2: Effect of the expansion ratio parameter on velocity f with Re =5 (Maple

RK45 solution) with Nt = Nb = a1 = b1 = c1= d1=0.1, Pe = Le = Lb = 0.5.

10, 5,0,5,10

17

Fig. 2 depicts the evolution in velocity f in the channel with variation in wall expansion

ratio ( ) with injection at the upper wall (Re = 5.0). With negative (upper wall

contracting) near the lower wall (static) and in the lower channel half space (0 0.5)

velocity is observed to be strongly reduced i.e. the flow is decelerated. Conversely with

positive (upper wall expanding) near the lower wall (static) and in the lower channel half

space (0 0.5) velocity is observed to be strongly elevated i.e. the flow is accelerated.

However in the upper channel half space (0.5 1.0) the contrary behavior is computed

i.e. the flow is decelerated with positive (upper wall expanding) whereas it is accelerated

with negative (upper wall contracting). Evidently the greatest acceleration occurs closer to

the lower plate for the expanding upper wall scenario since greater momentum transferred to

the body of the fluid and this in conjunction with the injection at the upper wall (permeation

Reynolds number, Re >0) aids flow development as far as into the lower channel half space.

The geometry we have studied is employed in certain microbial fuel cell (MFC) designs.

MFCs are biological fuel cells comprising a bio-electrochemical system which can be used to

harness bacterial (micro-organism) propulsion to generate a current externally. The designs in

which one wall is static and impervious and the other is stretchable have been shown to be

potentially conducive to developing asymmetric flow distributions of micro-organisms which

in turn can assist in increasing fuel cell efficiency. The extending/contracting nature of the

upper wall in fact is used to simulate lipid layers in these fuel cells which possess natural

elasticity. This allows them to adjust their state intelligently depending on the migration of

micro-organisms towards the upper wall. This also works as a “smart” design feature in such

fuel cells. Relevant works which elaborate on these structural wall features of next generation

fuel cells include Du et al. [49] who explore applications of such features in environmental

and bio-energy systems and Sun [50]. The presence of transpiration at the lower wall may

further be used as a mechanism to simulate super-capactive designs which utilize mass flux at

one or more fuel cell walls to improve energy yield in more complex energy-harvesting bio-

resource devices [51, 52]. In the present work we consider nanoscale-modified MFCs by

exploring the thermo-nano-bioconvection fluid dynamics of such systems and do not dwell on

efficiency (thermodynamic) aspects, although this constitutes a possible extension to the

work, which is being considered presently and which it is envisaged may be explored by

other researchers.

18

Figs. 3(a)-(c) display the variations in dimensionless temperature, nanoparticle volume

fraction and motile micro-organism density function, respectively, for different values of wall

expansion ratio ( ) and permeation Reynolds number (Re). Fig. 3a shows that for the

expanding upper wall case ( =0.5), there is a weak decrease in temperatures throughout the

channel span. Conversely for the contracting upper wall case, ( = -0.5) the temperature is

weakly increased. The stationary upper wall case ( =0) is observed to fall between the

expanding and contracting case profiles. With increasing permeation Reynolds number,

temperature is significantly elevated throughout the channel width. This is attributable to the

momentum boost in the flow associated with injected nanofluid via the upper wall. This

assists thermal diffusion and energizes the channel flow which will elevate biofuel cell

efficiency. Permeation Reynolds number embodies the relative significance of the inertia

effect compared to the viscous effect. For Re = 1 both inertial and viscous forces are of the

same order of magnitude. However for Re = 3, inertial force significantly exceeds viscous

forces and this in turn exacerbates the thermal diffusion in the nanofluid. In all cases,

maximum temperatures are attained at the upper (dynamic) wall and profiles exhibit a

monotonic ascent from the lower wall to converge smoothly at the upper wall. Fig. 3b

demonstrates that nano-particle species concentration (volume fraction) is also

enhanced with increasing Reynolds number from 1 to 3. The injection of nanofluid via the

upper wall therefore serves to elevate concentration of suspended nano-particles in the

channel and gradients of profiles are in fact sharper ascents than for the temperature field.

However the influence of the expansion parameter is the reverse of that computed for the

temperature field. For the expanding upper wall case ( =0.5), there is a substantial

enhancement in nanoparticle volume fraction (concentration), whereas for the contracting

upper wall case, ( = -0.5) the nanoparticle concentration is suppressed. The stationary upper

wall case ( =0) again lies between these other two cases. For both the temperature and

nano-particle concentration fields, the influence of Reynolds number and expansion

parameter is sustained across the entire channel. Fig. 3c shows that the micro-organism

density function, is increased with an expanding upper wall ( =0.5) whereas it is reduced

with a contracting upper wall ( = -0.5), only in the region near the lower static wall; further

from this zone there is a reversal in trends through the core flow one of the channel and for

much of the channel upper half space. The pattern is then reversed again as we approach the

upper wall. The peak micro-organism density values are localized closer to the lower static

wall for Re =3. They migrate further away for Re = 1 and are in fact only increased in

19

magnitude for the solitary case of ( =0.5). For = -0.5, 0 the magnitudes are markedly

lower than for Re = 3) although they are less asymmetrically distributed across the channel

span. Generally however the expansion ratio and Reynolds number exert a significant

influence on the magnitude of micro-organism density through the channel i.e. the

concentration of micro-organisms demonstrates notable sensitivity to a contracting/expanding

wall and also greater inertial forces (Reynolds number).

Figure 4(a)-(d) illustrate the response of effect of velocity (momentum) slip parameter 1a

and wall expansion ratio ( ) on dimensionless velocity, temperature, nanoparticle volume

fraction and motile micro-organism density profiles, respectively. Fig 4a shows that velocity

is weakly reduced near the lower wall of channel with positive i.e. slight deceleration is

induced with an expanding upper wall, whereas the flow is weakly accelerated with negative

i.e. contracting wall. Momentum slip is imposed only at the upper wall. It is absent from

the lower wall, as is apparent from the boundary condition f /(1)=a1f

//(1) in eqn. (19). With

an increase in 1a , velocity is accentuated in the lower channel half space whereas it is reduced

in the upper channel half space, most prominently at the upper wall. Therefore lower

momentum slip at the upper wall is observed to induce acceleration there whereas a strong

deceleration is generated near the lower static wall. Greater momentum slip retards the fluid

motion near and at the upper wall. Moreover, dragging of the fluid adjacent to the expanding

wall is partially transmitted into the fluid partially which induces a deceleration. This result is

in agreement with Raza et al. [36]. Fig. 4b reveals that dimensionless temperature decreases

continuously as momentum slip is elevated i.e. with greater values of 1a . All profiles exhibit

a monotonic ascent from the lower static wall to the upper wall. For an expanding wall

( =+0.5) temperature is decreased throughout the channel whereas for a contracting wall

( =-0.5) it is elevated. An expanding wall therefore cools the channel flow whereas a

contracting wall heats it. Fig 4c shows that a weak increase in nano-particle volume fraction

is caused with greater momentum slip and with a contracting wall ( =-0.5) whereas a

marginal decrease is induced with an expanding wall ( =+0.5). Micro-organism density

function is increased substantially near the lower static wall with an increase in momentum

slip from a1 =1 to a1=10. However this behavior is reversed as we approach the central zone

of the channel and progressively a marked reduction in micro-organism density function is

observed as we approach the upper wall. The presence of an expanding upper wall has a

similar effect to momentum slip i.e. it enhances micro-organism density function closer to the

lower wall implying that the concentration of motile micro-organisms is intensified near the

20

lower wall. However towards the upper wall and indeed in the channel core flow region, the

presence of an expanding wall is found to suppress micro-organism density function. The

contracting wall case ( =-0.5) causes the opposite effect and depresses micro-organism

density function near the lower static wall but elevates magnitudes towards the upper wall.

The clustering and distribution of micro-organisms is therefore profoundly modified

throughout the channel by the upper wall condition and also by momentum (hydrodynamic)

slip even though both conditions are only prescribed at the upper wall.

Figure 5 displays the influence of thermal slip parameter ( 1b ) and expansion ratio ( ) on

dimensionless temperature, (), in the channel. Thermal slip is also imposed solely at the

upper wall, as embodied in the boundary condition (1)=1+ b1 /

(1) in eqn. (19).

Increasing thermal slip is found to decrease temperatures in the lower channel half space

whereas it elevates temperatures in the upper channel half space i.e. in the region closer to the

upper wall. Maximum temperature computed arises at the upper wall. With increasing

thermal slip less heat is transmitted to the fluid in the lower channel half space whereas more

heat is conveyed in the upper channel half space. The zone near and at the upper wall is

therefore energized and heated. The influence of on temperature is primarily modelled via

the term, /Pr in eqn. (16). The presence of an expanding wall ( =+0.5) decreases

temperatures through the channel whereas a contracting wall manifests in a temperature

increase.

Figures 6(a)-(b) depict the impact of mass slip parameter on dimensionless concentration

() and motile micro-organism density function, () through the channel space. Fig. 6a

shows that as with momentum and thermal slip, nano-particle mass flip is imposed only at the

upper wall via the boundary condition (1)=1+ c1 / (1) in eqn. (19). Unlike the temperature

response in fig. 5, nano-particle volume fraction is observed to be enhanced throughout the

entire channel with increasing mass slip, as shown in fig. 6a. Peak values are achieved as

expected at the upper wall. The diffusion of nano-particles is therefore assisted in the

nanofluid with greater mass slip at the upper wall. Higher distributions of nanoparticles can

therefore be encouraged everywhere in the channel with the prescription of mass slip at the

upper wall, which has important implications for attaining better thermal performance

throughout the entire channel, not merely in localized zones. The influence of on nano-

particle concentration is analyzed via the term, /Pr Le in eqn. (17). Confirming earlier

21

graphs, we observe that an expanding wall ( =+0.5) suppresses nano-particle volume

fraction magnitudes whereas a contracting wall ( = -0.5) is responsible for enhancing them.

(a) (b)

(c)

Figure 3:Effect of the Re on the dimensionless (a) temperature (b) nanoparticle

volume fraction and (c) microorganism density function with prescribed

parameter values of 1 1 1 1 0.1, 10, 1,Lb 0.5a b c d Nt Nb Le Pe

0.5,0,0.5

Re 3

Re 1

Re 3

Re 1

0.5,0,0.5

0.5,0,0.5

Re 3

Re 1

22

(a) (b)

(c) (d)

Figure 4:Effect of the velocity slip parameter 1a on the dimensionless (a) velocity f (b)

temperature (c) nanoparticle volume fraction and (d) micro-organism density

function with values 1 1 1 0.1, 10, 1,Lb 0.5,Re 5b c d Nb Nt Le Pe .

0.5,0,0.5

1

1

10

1

a

a

0.5,0,0.5

1

1

10

1

a

a

0.5,0,0.5

1

1

10

1

a

a

0.5,0,0.5

1

1

10

1

a

a

23

Figure 5:Effect of the thermal slip parameter 1b on the dimensionless temperature with

parameter values of 1 1 1 1 0.1, 10, 1,Lb 0.5,Re 2a b c d Nt Nb Le Pe .

(a) (b)

Figure 6:Effect of the mass slip parameter 1c on the dimensionless a) nanoparticle volume

fraction and (b) microorganism density function with parameter values of

1 1 1 0.1, 10, 1,Lb 0.5,Re 1a b d Nb Nt Le Pe .

0.5,0,0.5

1

1

10

0.1

b

b

0.5,0,0.5

1

1

10

1

c

c

0.5,0,0.5

1

1

10

1

c

c

24

Figure 7: Effect of the micro-organism slip parameter d1 on the dimensionless micro-

organism density with 1 1 1 0.1, 10, 1,Lb 0.5,Re 3a b c Nb Nt Le Pe .

(a) (b)

0.5,0,0.5

1

1

10

1

d

d

0.1,0.3,0.5Nt

0.5

0.1

Nb

Nb

0.1,0.3,0.5Nt

0.5

0.1

Nb

Nb

25

(c)

Figure 8: Effect of the thermophoresis parameter Nt and Brownian motion parameter Nb on

the dimensionless (a) temperature (b) nanoparticle volume fraction and (c)

microorganism density function with parameter values of a1=

1 1 1 0.1, 1, 1,Lb 0.5,Re 3, 0.5b c d Le Pe .

Figure 9:Effect of the bioconvection Péclet number Pe and bioconvection Lewis number Lb

on the dimensionless micro-organism density function with

a1= 1 1 1 0.1, 1,Re 3, 0.5b c d Nb Nt Le

0.1,0.3,0.5Nt

0.5

0.1

Nb

Nb

1,3,5Lb

3

1

Pe

Pe

26

Therefore combining mass slip and contracting behaviour at the upper wall, overall achieves

the best performance and sustains the highest magnitudes in the channel for nano-particle

volume fraction. Fig. 6b shows that nano-particle mass slip also exerts a beneficial effect on

the motile micro-organism density function magnitudes. However the expansion ratio has the

reverse effect to that on the nano-particle volume fraction field. An expanding wall ( =+0.5)

strongly boosts the magnitudes of () through the entire channel whereas a contracting wall

results in a significant depression in () values everywhere. The influence of on micro-

organism density function is primarily simulated via the expansion ratio-micro-organism

coupling term, /PrLb in eqn. (18). Also evident is the fact that the peak micro-organism

density values migrate progressively from the lower static wall towards the central zone of

the channel as the upper wall changes from an expanding one to a contracting one, with the

stationary upper wall case intercalated between the other two cases. The distributions also

become more symmetric across the channel span. Therefore although an expanding upper

wall elevates substantially the micro-organism density magnitudes, these are confined to the

lower channel half space i.e. they are skewed towards the lower wall. Conversely the

contracting upper wall, although attaining lower magnitudes of micro-organism density,

nevertheless generates a more homogenous distribution across the channel which may impact

on microbial fuel cell efficiency. Irrespective of the value of mass slip or expansion ratio,

micro-organism densities are generally minimized at the upper wall. Therefore fuel cell

designers engaged in employing bioconvection in nanofluids are required to judiciously

deploy upper wall boundary conditions and also nano-particle slip to optimize micro-

organism replenishment and proliferation which impacts heavily on sustainable efficiency of

such systems. Further elaboration is given in Croze et al. [53]. In fig. 6a we have considered

the case Re = 1 implying that inertial forces are equivalent to viscous forces in the channel.

Further investigation, preferably experimental would shed greater light on the inter-play

between these forces and the susceptibility of micro-organism concentration and also nano-

particle diffusion to variation in the relative inertial and viscous effects.

Figure 7 illustrates the impact of micro-organism slip parameter (d1) on

dimensionless motile micro-organism density function, () through the channel space. This

slip factor is applied at the upper wall, in a similar fashion to thermal and nano-particle mass

slip via the boundary condition (1)=1+ d1 / (1) in eqn. (19). A more uniform response is

computed compared with the effect of nano-particle mass slip (which was studied earlier in

fig. 6b). Increasing micro-organism slip substantially elevates the micro-organism density

27

function primarily in the vicinity of the lower wall. The maximum, () magnitude arises

therefore in the lowest region of the lower channel half space. However as we approach the

lower central core region of the channel (0.2 0.3) there is a switch in behavior and

increasing micro-organism slip at the upper wall is observed to induce a marked decrease in

() magnitudes which is sustained throughout the upper channel half space. () values

progressively deplete until they attain a minimum at the upper wall. At the upper wall

therefore micro-organism density function is a maximum for lowest micro-organism slip

factor (d1=1).The presence of an expanding upper wall ( =+0.5) encourages motile micro-

organism density enhancement near the flower wall whereas a contracting upper wall (i.e.

expansion parameter = -0.5) suppresses values. We further note that very low values of

momentum, thermal and nano-particle mass slip (<<1) are imposed to allow the dominant

effect of micro-organism slip to be evaluated.

Figure 8 illustrates the collective influence of nano-particle thermophoresis parameter

( Nt ) and the Brownian motion parameter ( Nb ) on the dimensionless temperature,

nanoparticle volume fraction and motile micro-organism density function distributions across

the channel. The contracting upper wall case is examined ( = -0.5). Increasing Brownian

motion parameter physically correlates with smaller nanoparticle diameters, as elaborated in

Rana et al. [54] and Akbar et al. [55]. Smaller values of Nb corresponding to larger

nanoparticles, implies that surface area is reduced which in turn decreases thermal conduction

heat transfer to the fluid. This coupled with macro-convection manifests in fig. 8a, in a non-

trivial decrease in thermal energy imparted to the fluid and a concomitant fall in temperature,

(). The reverse effect is induced with larger values of Nb which correlate with smaller

nano-particles and enhanced thermal conduction leading to higher temperatures. An increase

in Nt also induces a rise in temperatures. Thermophoretic migration of nano-particles

encourages thermal diffusion in the regime and energizes the flow. This elevates

temperatures. Fig. 8b also shows that an elevation in nano-particle volume fraction, (),

accompanies an increase in Brownian motion parameter, Nb. In fact much greater

enhancement is achieves compared with the temperature field. Increasing thermophoresis

parameter, Nt, conversely suppresses the magnitudes of nano-particle volume fraction, ().

Nano-particle volume fraction is observed to ascend from zero values at the lower plate to

maximum magnitudes at the upper wall, and all profiles approach this latter limit smoothly.

Inspection of fig. 8c reveals that micro-organism density function profiles are distinctly

different from the temperature and nano-particle concentration distributions. While increasing

28

Nb values evidently promote diffusion of micro-organisms and achieve significant

enhancement in their magnitudes, conversely increasing thermophoretic parameter, Nt,

substantially depresses values. These trends are however restricted to the lower channel half

space. In the upper channel half space on the other hand, increasing thermophoresis is found

to elevate () magnitudes whereas increasing Brownian motion parameter increases ()

magnitudes. However peak values of micro-organism density function, () are localized in

the lower channel half space. The minimum value of () is observed to arise in the lower

channel half space at ~0.2, for Nt =0.5 and Nb = 0.1.

Figure 9 demonstrates the composite effects of bioconvection Lewis number and

bioconvection Péclet number on the dimensionless micro-organism density function. The

dimensionless micro-organism density magnitudes increase massively with in an increase Pe

whereas they are reduced significantly with an increase in Lb . In the computations, standard

Lewis number is prescribed as Le =1 which physically implies that thermal diffusivity of the

nanofluid and species diffusivity of the nano-particles are the same. Pe i.e. bioconvection

Péclet number embodies the ratio of advection rate of nano-particles to the diffusion rate. Pe

<10 is more realistic for simulating actual transport phenomena in bioconvection nanofluid

mechanics. Pe features only in the micro-organism density conservation eqn. (18) via the

coupling terms )( //// Pe which intimately connects the nano-particle concentration

(volume fraction) and micro-organism fields. These terms evidently have a pronounced

influence on the evolution of micro-organism density function in the channel. Bioconvection

Lewis number, Lb, signifies the ratio of dynamic viscosity of nanofluid to the species

diffusivity of micro-organisms, i.e. /Dn arises solely in the micro-organism density

conservation eqn. (18). It features in the velocity-micro-organism density coupling

term, /RePr fLb and furthermore in the expansion ratio-micro-organism coupling term,

/PrLb . The effect of Lb will therefore also be sustained in the momentum field i.e. eqn

(28). For Lb <1 the micro-organism diffusion rate exceeds viscous diffusion rate and vice

versa for Lb >1. With increasing Lb values there is a strong suppression in the micro-

organism density magnitudes, attributable to the decrease in micro-organism species

diffusivity in the nanofluid (or an increase in viscosity).

6. CONCLUSIONS

A two-dimensional unsteady laminar model for bioconvection nanofluid flow in a channel

with multiple upper wall slip effects and upper wall expansion/contraction has been studied.

29

The upper wall is porous whereas the lower wall is stationary and impermeable surface and

therefore injection is present at the upper wall. The governing partial differential equations

for momentum, energy, nano-particle species (volume fraction) and motile micro-organism

density conservation are transformed into a set of ordinary differential equations (ODEs)

using similarity variables. The resulting high order nonlinear ODEs are solved numerically

using Runge-Kutta-Felhberg integration scheme available in the RK45 shooting algorithm in

Maple software. Validation of solutions is achieved with a Nakamura tridiagonal second

order accurate finite difference scheme. Solutions are presented graphically and demonstrate

the significant influence of bioconvection Lewis number, bioconvection Peclét, momentum

slip, thermal slip, nano-particle mass slip, micro-organism slip, upper wall expansion ratio,

permeation Reynolds number, Brownian motion and thermophoretic parameter on all the

field variables. The principal deductions from the present investigation may be summarized

thus:

(i)Adding nanoparticles into base fluid containing suspended gyrotactic micro-organisms

serves to increase the nano-particle and micro-organism mass transfer rates and achieves a

stable mixture (suspension).

(ii) Increasing bioconvection Lewis number suppresses dimensionless micro-organism

density magnitudes whereas the reverse effect is induced with increasing bioconvection

Péclet number.

(iii) An increase in Brownian motion parameter elevates micro-organism density function

magnitudes whereas increasing thermophoretic parameter reduces them, in the lower channel

half space.

(iv) An increase in Brownian motion parameter also enhances nano-particle volume fraction

values, whereas greater thermophoresis parameter depresses values, throughout the channel

span. Increasing micro-organism slip is found to enhance micro-organism density function

mainly in the zone near the lower wall of the channel.

(v) An upper expanding wall decelerates the flow near the lower wall of channel whereas a

contracting wall weakly accelerates the flow.

(vi) Increasing momentum slip accelerates the flow in the lower channel half space whereas it

decelerates the flow in the upper channel half space, most prominently at the upper wall.

(vii) Increasing nano-particle mass slip at the upper wall elevates motile micro-organism

density function magnitudes.

(viii) Greater thermal slip reduces temperatures in the lower channel half space whereas it

enhances temperatures in the upper channel half space.

30

(ix) An expanding upper wall significantly elevates micro-organism density function

magnitudes through the entire channel whereas a contracting wall induces the opposite effect.

The present simulations have been confined to Newtonian bioconvection nanofluids. Future

studies will explore non-Newtonian models e.g. viscoelastic formulations [54], and will be

communicated shortly. Furthermore nano-particle geometric effects also warrant more

detailed analysis [56, 57] and these will also be explored. Additionally the current study has

been restricted to gyrotactic micro-organisms i.e. where compensating torques generated by

shear and gravity effects manifest in gyro-taxis which controls the orientation of up-

swimming micro-organisms via rotary motions. However many other taxes are prevalent and

can be exploited in microbial fuel cell design. These include rheotaxis (where the dominant

response is to shear in the ambient flow. Certain bacterial swimmers contain magnetic

particles (magnetosomes) and these cause a strong bias of the swimming motion along

magnetic field lines and are known as magnetotaxis. Gravitaxis is a gravitational-driven or

acceleration-driven taxis and for certain photosynthetic algae predominant propulsion is

generally vertically upwards. Other significant taxes are photo-taxis and chemo-taxis in which

the dominant effect is a swimming towards more intense light and chemical gradients,

respectively. All these taxes are in fact relevant to microbial fuel cell (MFC) design and

infact combinations may also be exploited. To simulate these taxes the hydrodynamic models

must be modified to emphasize the particular taxis. This involves detailed micro-

hydrodynamic analysis and modifications in hydrodynamic torque formulation. The direction

in which a micro-organism swims, is defined at any instant by the balance of viscous and

external torques on the cell. These external torques will be gravitational for gyrotaxis and will

be modified for other external taxes e.g. phototaxis or chemo-taxis. This allows the model to

be specific for a particular type of directional swimming behavior (light, oxygen

concentration, etc.). Relevant photo-tactic and chemo-tactic models have been communicated

in this regard by Ghorai and Hill [58] and Yanaoka et al. [59] and are currently being

explored.

ACKNOWLEDGEMENTS:

The authors acknowledge financial support from Universiti Sains Malaysia, RU Grant

1001/PMATHS/81125. The authors are extremely grateful to both Reviewers for their

excellent and informative comments which have certainly served to clarify and improve the

present work. O. Anwar Bég expresses his deepest gratitude to Tazzy Bég.

31

REFERENCES

[1] Hedayati F and Domairry G., Nanoparticle migration effects on fully developed forced

convection of TiO 2 – water nanofluid in a parallel plate microchannel, Particuology 24 96–

107.

[2] Wu L., Mass transfer induced slip effect on viscous gas flows above a shrinking /

stretching sheet Int. J. Heat Mass Transf. 93 17–22 (2016).

[3] Mosayebidorcheh S., Analytical investigation of the micropolar flow through a porous

channel with changing walls J. Mol. Liq. 196 113–9 (2014).

[4] Odelu O. and Naresh Kumar N., Slip-flow and heat transfer of chemically reacting

micropolar fluid through expanding or contracting walls with Hall and ion slip currents Ain

Shams Eng. J. (2015). DOI:10.1016/J.ASEJ.2015.09.011

[5] Adesanya S O, Free convective flow of heat generating fluid through a porous vertical

channel with velocity slip and temperature jump Ain Shams Eng. J. 1045–52 (2014).

[6] Uchida S and Aoki H., Unsteady flows in a semi-infinite contracting or expanding pipe J.

Fluid Mech. 82 371–87 (1977).

[7] N.M. Bujurke, V.S. Madalli, B.G. Mulimani, Long series analysis of laminar flow

through parallel and uniformly porous walls of different permeability, Computer Methods in

Applied Mechanics and Engineering, 160, 39-56 (1998).

[8] M.Goto and S. Uchida, Unsteady flows in a semi-infinite expanding pipe with injection

through wall, J. Japan Society for Aeronautical and Space Sciences, 33, 14-27 (1990).

[9] Hatami M, Sheikholeslami M and Ganji D D., Nanofluid flow and heat transfer in an

asymmetric porous channel with expanding or contracting wall, J. Mol. Liq. 195 230–9

(2014).

[10] Boutros Y Z, Abd-el-Malek M B, Badran N A and Hassan H S., Lie-group method for

unsteady flows in a semi-infinite expanding or contracting pipe with injection or suction

through a porous wall J. Comput. Appl. Math. 197 465–94 (2006).

[11] Xinhui S, Liancun Z, Xinxin Z and Jianhong Y, Homotopy analysis method for the heat

transfer in a asymmetric porous channel with an expanding or contracting wall Appl. Math.

Model. 35 4321–4329 (2011).

[12] Si X, Zheng L, Zhang X, Li M, Yang J and Chao Y., Multiple solutions for the laminar

flow in a porous pipe with suction at slowly expanding or contracting wall Appl. Math.

Comput. 218 3515–21 (2011).

[13] Ahmed N, Mohyud-Din S T and Hassan S M Flow and heat transfer of nanofluid in an

asymmetric channel with expanding and contracting walls suspended by carbon nanotubes: A

numerical investigation Aerosp. Sci. Technol. 48 53–60 (2016).

[14] Darvishi M T, Khani F, Awad F G, Khidir A A and Sibanda P., Numerical investigation

of the flow of a micropolar fluid through a porous channel with expanding or contracting

walls Propuls. Power Res. 3 133–142 (2014).

32

[15] S.U. Choi, Enhancing thermal conductivity of fluids with nanoparticles, Development

and Applications of Non-Newtonian Flow, FED-Vol. 231/MD-Vol. 66, ASME, 99–105

(1995).

[16] J.A. Eastman, S.R. Phillpot, S.U.S. Choi, P. Keblinski, Thermal transport in nanofluids,

Ann. Rev. Mater. Res. 34, 219-146 (2004).

[17] Khairul M A, Shah K, Doroodchi E, Azizian R and Moghtaderi B., Effects of surfactant

on stability and thermo-physical properties of metal oxide nanofluids Int. J. Heat Mass

Transf. 98 778–87 (2016).

[18] O. Anwar Bég and D. Tripathi, Peristaltic pumping of nanofluids, Chapter 3, pages 69-

96, Modeling and Simulation Methods and Applications, Eds (S. K. Basu, N. Kumar),

Springer, Berlin, Germany (2014).

[19] Kakaç S and Pramuanjaroenkij A., Review of convective heat transfer enhancement with

nanofluids Int. J. Heat Mass Transf. 52 3187–96 (2009).

[20] Hamad M A A and Ferdows M., Similarity solution of boundary layer stagnation-point

flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption

/ generation and suction / blowing : A Lie group analysis Commun. Nonlinear Sci. Numer.

Simul. 17 132–40 (2012).

[21] Pedley T J, Hill N. and Kessler J. O. , The growth of bioconvection patterns in a uniform

suspension of gyrotactic micro-organisms. J. Fluid Mech. 195 223–37 (1988).

[22] Uddin M.J., Kabir M N and O. Anwar Bég, Computational investigation of Stefan

blowing and multiple-slip effects on buoyancy-driven bioconvection nanofluid flow with

microorganisms, International Journal of Heat and Mass Transfer. 95 116–30 (2016).

[23] A. Raees, H. Xu, Q. sun and I. Pop, Mixed convection in gravity-driven nano-liquid film

containing both nanoparticles and gyrotactic microorganisms, Applied Mathematics and

Mechanics, 36, 163-178 (2015).

[24] Raees A, Xu H and Liao S., Unsteady mixed nano-bioconvection flow in a horizontal

channel with its upper plate expanding or contracting, Int. J. Heat and Mass Transfer. 86

174–82 (2015).

[25] Md Faisal Md Basir, M.J. Uddin, A. I. Md. Ismail and O. Anwar Bég, Unsteady bio-

nanofluid slip flow over a stretching cylinder with bioconvection Schmidt and Péclet number

effects, AIP Advances, 6, 055316-1 - 055316-15 (2016).

[26] J. Buongiorno Convective transport in nanofluids, ASME J. Heat Transfer, 128, 240–250

(2006).

[27] A. Malvandi, S.A. Moshizi, Elias Ghadam Soltani, D.D. Ganji, Modified Buongiorno’s

model for fully developed mixed convection flow of nanofluids in a vertical annular pipe,

Computers & Fluids, 89, 124–132 (2014).

33

[28] Hang Xu, Tao Fan, Ioan Pop, Analysis of mixed convection flow of a nanofluid in a

vertical channel with the Buongiorno mathematical model, International Communications in

Heat and Mass Transfer, 44, 15–22 (2013).

[29] M. A. Sheremet, Ioan Pop, Free convection in a triangular cavity filled with a porous

medium saturated by a nanofluid: Buongiorno’s mathematical model", International Journal

of Numerical Methods for Heat & Fluid Flow, 25, 1138 – 1161 (2015).

[30] N. S. Akbar, D. Tripathi, Z.Khan and O. Anwar Bég, A numerical study of

magnetohydrodynamic transport of nanofluids from a vertical stretching sheet with

exponential temperature-dependent viscosity and buoyancy effects, Chemical Physics Letters,

661, 20-30 (2016).

[31] M.J. Uddin, O. Anwar Bég and A.I. Ismail, Radiative-convective nanofluid flow past a

stretching/shrinking sheet with slip effects, AIAA J. Thermophysics Heat Transfer, 29, 3, 513-

523 (2015).

[32] M. J. Uddin, W.A. Khan, A.I.Md. Ismail, O. Anwar Bég, Computational study of three-

dimensional stagnation point (Buongiornio) nanofluid bio-convection flow on a moving

surface with anisotropic slip and thermal jump effects, ASME J. Heat Transfer (USA) (2016).

DOI: 10.1115/1.4033581 (8 pages).

[33] Kuznetsov A V. and Nield D. A., The Cheng-Minkowycz problem for natural

convective boundary layer flow in a porous medium saturated by a nanofluid: A revised

model Int. J. Heat Mass Transf., 65, 682–5 (2013).

[34] Dauenhauer E C and Majdalani J., Exact self-similarity solution of the Navier-Stokes

equations for a porous channel with orthogonally moving walls, Phys. Fluids 15 1485–95

(2003).

[35] Cao L, Si X and Zheng L., The flow of a micropolar fluid through a porous expanding

channel: A Lie group analysis Appl. Math. Comput. 270 242–50 (2015).

[36] Raza J, Rohni A M, Omar Z and Awais M., Heat and mass transfer analysis of MHD

nanofluid flow in a rotating channel with slip effects J. Mol. Liq. 219 703–8 (2016).

[37] Pedley, T. J., Kessler, J. O. 1990. A new continuum model for suspensions of gyrotactic

micro-organisms. J. Fluid Mech. 212: 155-82.

[38] Hart, A., Edwards, C 1987. Buoyant density fluctuations during the cell cycle of Bacillus

sub/ilis. Arch. Microbiol. 147: 68-72.

[39] Md. Jashim Uddin, O. Anwar Bég and Ahmad Izani Md. Ismail, Mathematical

modelling of radiative hydromagnetic thermo-solutal nanofluid convection slip flow in

saturated porous media, Math. Prob. Engineering, 2014, Article ID 179172, 11 pages (2014).

[40] O. Anwar Bég, M. J. Uddin and W.A. Khan, Bioconvective non-Newtonian nanofluid

transport in porous media containing micro-organisms in a moving free stream, J. Mechanics

Medicine Biology, 15, 1550071.1-1550071.20 (2015).

34

[41] O. Anwar Bég and O.D. Makinde, Viscoelastic flow and species transfer in a Darcian

high-permeability channel, Petroleum Science and Engineering, 76, 93–99 (2011).

[42] Nakamura, S., Iterative finite difference schemes for similar and non-similar boundary

layer equations, Adv. Eng. Software, 21, 123–130 (1994).

[43] Bég, O. Anwar, NANONAK- A finite difference code for nanofluid convection

problems of the boundary layer type, Technical Report, NANO-C/5-1, 124pp, Gort

Engovation, Bradford, England and Narvik, Norway, UK, August (2013).

[44] Bég, O. Anwar, J. Zueco, M. Norouzi, M. Davoodi, A. A. Joneidi, Assma F. Elsayed,

Network and Nakamura tridiagonal computational simulation of electrically-conducting

biopolymer micro-morphic transport phenomena, Computers in Biology and Medicine, 44,

44–56 (2014).

[45] Bég, O. Anwar, T. A. Bég, H. S. Takhar, A. Raptis, Mathematical and numerical

modeling of non-Newtonian thermo-hydrodynamic flow in non-Darcy porous media, Int. J.

Fluid Mech. Res., 31, 1–12 (2004).

[46] Gorla, R.S.R. and S. Nakamura, Combined convection from a rotating cone to

micropolar fluids, Math. Modelling Sci. Comput., 2, 1249–1254 (1993).

[47] Nakamura, S., Applied Numerical Methods and Software, Prentice-Hall New Jersey,

USA (1995).

[48] Bég, O. Anwar, Numerical methods for multi-physical magnetohydrodynamics, Chapter

1, pp. 1-112, New Developments in Hydrodynamics Research, M. J. Ibragimov and M. A.

Anisimov, Eds., Nova Science, New York, September (2012).

[49] Z. Du, H. Li, and T. Gu, A state of the art review on microbial fuel cells: A promising

technology for wastewater treatment and bioenergy, Biotechnology Advances, 25, 464-482

(2007).

[50] Y. Sun, Electricity generation and microbial community changes in microbial fuel cells

packed with different anodic materials, Bioresource Technology, 10, 10886-10891 (2011).

[51] H. Wang, J-D. Park and Z. J. Ren, Practical energy harvesting for microbial fuel cells: a

review, Environ. Sci. Technol., 49 (6), 3267–3277 (2015).

[52] J. Houghton, C. Santoro, F. Soavi, A. Serov, I. Ieropoulos, C. Arbizzani, P. Atanassov,

Supercapacitive microbial fuel cell: Characterization and analysis for improved charge

storage/delivery performance, Bioresource Technology, 218, 552-560 (2016).

[53] O.A. Croze, E. Ashraf and M. Bees, Sheared bioconvection in a horizontal tube,

Physical Biology, 7(4):046001 (2010).

[54] P. Rana, R. Bhargava, O. Anwar Bég and A. Kadir, Finite element analysis of

viscoelastic nanofluid flow with energy dissipation and internal heat source/sink effects, Int.

J. Applied Computational Mathematics, 1-27 (2016). DOI 10.1007/s40819-016-0184-5.

35

[55] N. S. Akbar, O. Anwar Bég, Z. H. Khan, Magneto-nanofluid flow with heat transfer past

a stretching surface for the new heat flux model using numerical approach, Int. J. Num. Meth.

Heat Fluid Flow (2016). In press

[56] N. S. Akbar, D. Tripathi and O. Anwar Bég, Modelling nanoparticle geometry effects

on peristaltic pumping of medical magnetohydrodynamic nanofluids with heat transfer, J.

Mechanics in Medicine and Biology, 16 (2) 1650088.1-1650088.20. DOI:

10.1142/S0219519416500883 (2015).

[57] A. Askounis et al., Effect of particle geometry on triple line motion of nano-fluid drops

and deposit nano-structuring, Advances in Colloid and Interface Science, 222, 44–57 (2015).

[58]S. Ghorai and N.A. Hill, Penetrative phototactic bioconvection, Phys. Fluids, 17, 074101

(2005).

[59] H. Yanaoka, T. Inamura, K. Suzuki, Numerical analysis of bioconvection generated by

chemotactic bacteria, Japan Soc. Mech. Eng. J. Fluid Science and Technology, 4, 536-545

(2009).